Definition: A linear transformation (T: V \to W) is invertible if there exists a linear transformation (S: W \to V) such that: [S \circ T = I_V \quad \text{and} \quad T \circ S = I_W]
where (I_V) and (I_W) are the identity transformations on (V) and (W) respectively.
For finite-dimensional spaces, the following are equivalent:
(T) is invertible
(T) is one-to-one and onto
(\ker(T) = {\mathbf{0}}) and (\text{range}(T) = W)
(\dim(V) = \dim(W)) and (T) is one-to-one (or onto)
The matrix of (T) (relative to any bases) is nonsingular (has nonzero determinant)
Example: Rotation by any angle (\theta) in (\mathbb{R}^2) is invertible; its inverse is rotation by (-\theta).
**Definition:** A linear transformation \(T: V \to W\) is *invertible* if there exists a linear transformation \(S: W \to V\) such that:
\[S \circ T = I_V \quad \text{and} \quad T \circ S = I_W\]
where \(I_V\) and \(I_W\) are the identity transformations on \(V\) and \(W\) respectively.
For finite-dimensional spaces, the following are equivalent:
- \(T\) is invertible
- \(T\) is one-to-one and onto
- \(\ker(T) = \{\mathbf{0}\}\) and \(\text{range}(T) = W\)
- \(\dim(V) = \dim(W)\) and \(T\) is one-to-one (or onto)
- The matrix of \(T\) (relative to any bases) is nonsingular (has nonzero determinant)
**Example:** Rotation by any angle \(\theta\) in \(\mathbb{R}^2\) is invertible; its inverse is rotation by \(-\theta\).